On Pauli and the interconnectedness of all things

Do all the electrons in the universe really move about imperceptibly when Brian Cox rubs a diamond, and is it anything to do with the Pauli exclusion principle?

There's been some argument on twitter and elsewhere as to whether Brian Cox made a mistake in his excellent "Night with the Stars" earlier this year. Apart from some below-the-line nonsense in blog comments, it's an interesting physics discussion and it made me refine my own thoughts about quantum mechanics a bit. So I feel like writing about it. However, this article is going to be more aimed at physicists than usual, so if I assume too much prior knowledge, please accept my apologies and come back for the next one.

Here's the clip in contention:

The controversy is about whether all the electrons in the universe really move energy levels imperceptibly when Brian heats the diamond, whether it's instantaneous, and whether it is anything to do with the Pauli exclusion principle. The Pauli exclusion principle says that no two fermions can be in the same quantum state. Lily wrote a bit about it here.

Brian's main response to criticism so far has been to point to these lecture notes by his co-author Jeff Forshaw.

Now, declaration of interest, Brian and Jeff are both old friends of mine, and I even starred briefly in "Night of the Stars" as "elbow behind Jonathan Ross's head". I have never met Sean, though I have read some of his work (and used his links) and I have a lot of respect for him. Anyway, this is about physics, not about taking sides in a celebrity scientist face-off.

So... Do all the electrons in the universe shuffle about when Brian rubs his diamond? And is that anything to do with the Pauli exclusion principle?

Interconnection

If you treat each electron as though it sits in a "potential well" (created by the electrostatic field of the atomic nucleus), a good place to start is by thinking of a universe full of atoms as though it were a bunch of potential wells.

Next you have to solve this multiple-potential-well problem. Jeff's notes solve the Schroedinger equation for two wells as an example, and show that whereas for an infinite potential you'd have two spatially-localised energy levels with identical energies, for a finite potential you have two non-localised energy levels with very slightly different energies.

The animation at the end of the notes illustrates that an electron that starts off localised to a single atom is no longer an exact eigenstate in a multiple-atom situation. So for example in a system with two protons and only one electron, which starts with the electron bound to one of the protons, the electron would over time oscillate between the two protons. In principle no matter how far apart they are, though the oscillation period may be longer than the age of the universe. A long wait before you get invited to that sort of party.

This means that, since in nature no "potential well" is really infinite, over enough time no electron is truly localised to a particular atom. So in principle one has to treat the potential of the whole universe, all the atoms, as a single system (a single Hamiltonian). All agree on this, as far as I can tell.

This already means that saying "it's in a different place" is not sufficient reason to say of an electron "it's in a different quantum state".

Also it sort of already concedes Brian's point about the interconnectedness of all electrons. But as Sean points out, it is nothing to do with the Pauli exclusion principle.

Pauli Exclusion?

In fact even waving the diamond around the lecture theatre changes the potential between it and the other electrons in the universe; Sean also pointed this out. The Hamiltonian changes. So what..? Still nothing much to do with the Pauli exclusion principle.

The splitting (breaking of the degeneracy) in the energy levels shown in Jeff's notes allows the Pauli exclusion to work, but spin is neglected in Jeff's discussion, and as far as I can see the whole argument in there works for bosons just as well as fermions. Bosons aren't subject to an exclusion principle.

The Pauli exclusion principle does mean that you can put only one electron in each of the different definite energy levels1. But none of them are localised to a particular atom/well. And extrapolated to Brian's diamond, and the universe, this means that e.g. for a zillion carbon atoms, there are a zillion nearly-but-not-quite-equal-energy levels in the universe. And all of them involve all zillion atoms.

Now, if you heat the diamond, you can raise the energy level of some electrons2. Presumably some other electrons can then drop down to the vacated levels and so there is some potential impact. In any case, the excited state also respects the Pauli exclusion principle and that involves all the electrons in the universe.

So there's an interconnectedness in principle, though it is experimentally irrelevant because the energy shifts are minute and the oscillation periods are huge, and while it is non-local, so I guess in a sense instantaneous (whatever than might mean), you can't use it to transmit information so it doesn't violate causality or relativity.

Then Pauli... the splitting of energy levels demonstrated in Jeff's notes is not caused by the Pauli exclusion principle. But it is necessary to accommodate the Pauli exclusion principle, and bind atoms together into molecules. And the exclusion principle does then provide an additional (anti)correlation between all the electrons in the universe.

Practically speaking...

The whole "interconnectedness" business is experimentally unobservable at long distances. So that aspect is of limited interest except that it's a curiosity built in to the best physical theory we have. Whether you want to call it "wonder" or "**it" is up to you I suppose, but it's there in quantum mechanics.

To me the really interesting thing is how the energy levels in atoms change and mix as you bring atoms closer together. At long distances it becomes irrelevant (look how many decimal places place Jeff needs to go to before you can see the splitting in his second example!), but when atoms get near it's vital, and in the end, and coupled with the Pauli exclusion principle, it leads to molecular bonding, chemistry, life. And Life & Physics. And personally I understand this more clearly now than I did before.

I'd call it a high-score draw, myself, and a good game. If I have got stuff wrong here, I fully expect to be educated further. Though I don't think I have.

1 two if you count the two spin states but that's not the point. And in fact they'll be in an anti-symmetrised superposition, as Sean said in a comment to his own blog.

2 between the imperceptibly close nearly-ground-state levels, mostly, I guess, though there's a finite, exponentially suppressed, probability of bigger jumps.

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